Persistence of steady flows of a two-dimensional perfect fluid in deformed domains

Wirosoetisno, Djoko; Vanneste, Jacques

Abstract

The robustness of steady solutions of the Euler equations for two-dimensional, incompressible and inviscid fluids is examined by studying their persistence for small deformations of the fluid-domain boundary. Starting with a given steady flow in a domain D0, we consider the class of flows in a deformed domain D that can be obtained by rearrangement of the vorticity by an area-preserving diffeomorphism. We provide conditions for the existence and (local) uniqueness of a steady flow in this class when D is sufficiently close to D0 in Ck,α, k ≥ 3 and 0 < α < 1. We consider first the case where D0 is a periodic channel and the flow in D0 is parallel and show that the existence and uniqueness are ensured for flows with non-vanishing velocity. We then consider the case of smooth steady flows in a more general domain D0. The persistence of the stability of steady flows established using the energy–Casimir or, in the parallel case, the energy–Casimir–momentum method, is also examined. A numerical example of a steady flow obtained by deforming a parallel flow is presented.

Citation

Wirosoetisno, D., & Vanneste, J. (2005). Persistence of steady flows of a two-dimensional perfect fluid in deformed domains. Nonlinearity, 18(6), 2657-2680. https://doi.org/10.1088/0951-7715/18/6/013

• Journal Article Type: Article
• Publication Date: 2005-11
• Deposit Date: Feb 29, 2008
• Journal: Nonlinearity
• Print ISSN: 0951-7715
• Electronic ISSN: 1361-6544
• Publisher: IOP Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 18
• Issue: 6
• Pages: 2657-2680
• DOI: https://doi.org/10.1088/0951-7715/18/6/013